Method and non-invasive device for focusing acoustic waves

ABSTRACT

The invention concerns a method for focusing acoustic waves useful for obtaining an image of a field to be observed in a dissipative heterogeneous medium ( 2, 3 ) around which acoustic transducers (T 1 –Tn, T′ 1 –T′m) forming an imaging network and a target network. The method consists in following a training step during which pulse responses from the medium are measured between each transducer (Ti) of the imaging network ( 5 ) and several transducers (Tj) of the target network ( 6 ); deducing therefrom reference signals to be emitted by the transducers of the imaging network to produce a focused acoustic pulse in each transducer of the target network, then cumulatively, in determining reference signals to be emitted to focus an acoustic pulse on predetermined points in the medium. Said reference signals are stored and used subsequently to generate an acoustic image of the medium.

FIELD OF THE DISCLOSURE

This application is a 371 of PCT/FR01/03208 filed on Oct. 17, 2001.

The present invention relates to noninvasive methods and devices forfocusing acoustic waves, in particular ultrasound waves.

BACKGROUND OF THE DISCLOSURE

More particularly, the invention relates to a noninvasive method forfocusing acoustic waves in a dissipative heterogeneous medium comprisinga substantially homogeneous medium (for example the brain) surrounded atleast partially by a dissipative aberrating layer (for example theskull) which generates aberrations in the propagation of the acousticwaves, the acoustic waves being emitted from outside the aberratinglayer and focused in the substantially homogeneous medium.

The methods of this type which are commonly used do not make it possibleto obtain good focusing of the acoustic waves inside the medium, and,when these methods are used in imaging applications, they therefore donot make it possible to obtain a good resolution and a good imagecontrast when the propagation aberrations are significant, for examplewhen echography of the brain is being carried out from outside theskull.

It is, in particular, an object of the present invention to overcomethis drawback.

SUMMARY OF THE DISCLOSURE

To this end, according to the invention, a focusing method of the typein question is characterized in that it includes the following steps:

(a) an initial positioning step during which a number t greater than 2of acoustic transducers are fixed in predetermined positions outside theaberrating layer, these transducers being in (direct or indirect)contact with said aberrating layer and forming at least:

-   -   an imaging array which combines a number n between 1 and t of        said transducers,    -   and a target array which combines a number m between 1 and t of        said transducers (these two arrays may be entirely separate, or        include certain common transducers, or alternatively each        include all of the aforementioned transducers),

(b) a learning step itself comprising the following substeps:

(b1) a substep of learning to focus the imaging array on the targetarray, during which substep:

(b11) impulse responses hri(t) of the dissipative heterogeneous mediumare determined, respectively between each transducer i of the imagingarray and a plurality of focusing points r lying on the aberrating layerin respective correspondence with transducers of the target array (thisdetermination may be carried out by direct measurement if thetransducers of the target array are made to emit acoustic pulses, oroptionally by measurement and calculation if the transducers of thetarget array are made to emit acoustic signals other than pulses; thevalues measured and/or calculated in this way may then optionally becorrected by digital backpropagation in order to simulate transducerslying directly in contact with the aberrating layer if the transducersare not in direct contact with the aberrating layer), these impulseresponses being stored in digital form with a certain time samplingwhich determines a number p of frequency components of the impulseresponse, with respective frequencies ωk, i being an index between 1 andn which designates a transducer of the imaging array, r being an indexbetween 1 and m which designates a focusing point corresponding to atransducer of the target array and k being an index between 1 and pwhich designates a frequency component,

(b12) on the basis of these impulse responses, for each focusing point rcorresponding to a transducer of the target array, a set of n referencetime signals e′i(t,r) is calculated, i varying between 1 and n, suchthat if the aberrating wall were removed in the vicinity of the focusingpoint r, the emission of these reference signals by the varioustransducers i of the imaging array would generate a predetermined signal(for example an acoustic pulse) focused on the focusing point r,

(b2) a substep of focusing at a number R of predetermined focusingpoints lying in the substantially homogeneous medium, with indices qbetween m+1 and m+R, this substep consisting in determining for each ofthese focusing points q, moving step-by-step away from the focusingpoints 1 to m corresponding to the transducers of the target array,reference signals e′i(t,q) to be emitted by the various transducers i ofthe imaging array in order to generate a pulse focused on said focusingpoint q, the reference signals e′i(t,q) being determined for eachfocusing point q by proceeding as follows:

(b21) a first estimate of e′i(t,q), for i ranging from 1 to n, iscalculated on the basis of at least one reference signal e′i(t,q0), q0being the index of at least one focusing point close to the focusingpoint q for which the reference signal has already been determined, thiscalculation being performed by using an average speed of the acousticwaves in the substantially homogeneous medium (2),

(b22) the transducers of the imaging array are made to emit, byiterations, the estimates previously obtained of the reference signalse′i(t,q), then signals s_(i)(t,q) back-scattered by the dissipativeheterogeneous medium are picked up with the same transducers, then thesereference signals e′i(t,q) are modified for the next iteration in thefollowing way:e_(i)′(t)→α_(i)(q).e_(i)′(t−τ_(i)(q))where the values α_(i)(q) and τ_(i)(q) are a corrective amplitude factorand a corrective delay, which are calculated so as to maximize acoherence criterion C between said back-scattered signals, saiditerations being stopped when the criterion C reaches a predeterminedthreshold,

(b3) the reference signals e′i(t,q) are stored, at least for q betweenm+1 and m+R,

(c) and a focusing step during which, for at least one of said focusingpoints q, the transducers of the imaging array are made to emit saidreference signals e′i(t,q), i being an index between 1 and n designatinga transducer of the imaging array.

By virtue of these provisions, the propagation aberrations of theacoustic waves in the dissipative heterogeneous medium are overcome andvery precise focusing is obtained, which may in particular make itpossible to obtain reliable and precise echography of a field to beobserved through the aberrating layer by back-scattering, when acousticwaves focused on different points of the field to be observed aresuccessively emitted and the back-scattered acoustic waves are pickedup.

This precise focusing may also be used in applications other thanechography, in particular:

-   -   Doppler color imaging,    -   elastographic imaging methods, such as the one described in        document WO-A-00/55 616,    -   nonlinear imaging methods (“harmonic imaging”),    -   methods of treatment by localized destruction of a part of the        dissipative heterogeneous medium, in particular by hyperthermia,    -   methods for measuring optical absorption parameters of tissues        with activation by ultrasound, etc.

In preferred embodiments of the invention, one and/or other of thefollowing provisions may optionally be implemented:

-   -   during substep (b11), when at least certain transducers (of the        target array and/or the imaging array) are in contact with an        intermediate homogeneous medium (for example a gel) which is        itself in contact with the aberrating layer, the impulse        responses hri(t) are corrected by digital backpropagation in        order to simulate transducers lying directly in contact with the        aberrating layer;    -   substep (b12) itself includes the following substeps:

(b121) p transfer matrices H(ωk)=[Hri(ωk)] are determined, i rangingfrom 1 to n and r ranging from 1 to m, where Hri(ωk) is the value, atthe frequency ωk, of the Fourier transform of the impulse responsehri(t),

(b122) for each focusing point r corresponding to a transducer of thetarget array, n components Ei(ωk,r) are determined, i varying between 1and n, such that F(ωk,r)=H(ωk).E(ωk,r), where E(ωk,r)=[Ei(ωk,r)] is avector with n components, F(ωk,r) is a vector with m componentsFl(ωk,r), l varying between 1 and m, these m components Fl(ωk,r)corresponding to a desired focusing of the acoustic waves at thefrequency ωk on the focusing point r corresponding to a transducer ofthe target array,

(b123) for each focusing point r corresponding to a transducer of thetarget array, a vector of n time signals e(t,r)=[ei(t,r)] is deducedtherefrom, i varying between 1 and n, where

${e_{i}\left( {t,r} \right)} = {\sum\limits_{k = 1}^{P}{{{Ei}\left( {{\omega\; k},r} \right)} \cdot {\mathbb{e}}^{{j\;\omega\; k},t}}}$in complex notation, these signals ei(t,r) being adapted so that theemission of them respectively by the various transducers i of theimaging array generates an acoustic pulse focused on the focusing pointr of the target array,

(b124) a substep of correcting the aberrations generated by theaberrating layer between the substantially homogeneous medium and eachtarget transducer r, these aberrations being estimated on the basis ofthe measurements carried out previously, the aberrations estimated inthis way being used to calculate said reference time signals e′i(t,r);

-   -   p matrices H⁻¹(ωk) are calculated during substep (b122),        respectively by regularization and inversion of the transfer        matrices H(ωk), and the vector E(ωk,r) is calculated for each        transducer r of the target array by the formula:        E(ωk,r)=H ⁻¹(ωk).F(ωk,j);    -   during step (b122), the components Fl(ωk,r) of the vector        F(ωk,r) corresponding to the spatial distribution of the desired        field at the frequency ωk are equal to 0 for l≠r and to 1 for        l=r;    -   during substep (b124), the aberrating wall in the vicinity of        each focusing point r corresponding to a transducer of the        target array is assimilated to a filter, which has a finite        impulse response and is defined at each frequency ωk by an        amplitude Gr(ωk) and a phase φ_(r)(ωk), substep (b124) itself        including the following substeps:

(b1241) for each frequency ωk, the amplitude Gr(ωk) and the phaseφ_(r)(ωk) are calculated on the basis either of the signals ei(t,r) orof the vectors E(ωk,r),

(b1242) p corrected transfer matrices H′(ωk)=[H′ji(ωk)] are calculated,where

${{H_{ji}^{\prime}\left( \omega_{k}\; \right)} = {{{H_{ji}\left( \omega_{k} \right)} \cdot \frac{1}{G_{j}\left( \omega_{k} \right)}}{\mathbb{e}}^{- {{j\phi}_{j}{(\omega_{k})}}}}},$

(b1243) for each transducer r of the target array, n componentsE′i(ωk,r) are determined, i varying between 1 and n, such thatF(ωk,r)=H′(ωk).E′(ωk,r), where E′(ωk,r)=[Ei(ωk,r)] is a vector with ncomponents, F(ωk,r) is a vector with m components Fl(ωk,r), l varyingbetween 1 and m, these m components Fl(ωk,r) corresponding to a desiredfocusing of the acoustic waves at the frequency ωk on the focusing pointr corresponding to a transducer of the target array,

(b1244) for each focusing point r corresponding to a transducer of thetarget array, a vector of n reference time signals e′(t,r)=[e′i(t,r)] isdeduced therefrom, i varying between 1 and n, where

${{e_{i}^{\prime}\left( {t,r} \right)} = {\sum\limits_{k = 1}^{n}{E^{\prime}{{i\left( {{\omega\; k},r}\; \right)} \cdot {\mathbb{e}}^{{j\;\omega\; k},t}}}}}\mspace{14mu}$in complex notation;

-   -   during substep (b2141), the amplitude Gr(ωk) and the phase        φ_(r)(ωk) are calculated as follows:

$\begin{matrix}{{{Gr}\left( \omega_{k} \right)} = \frac{\sqrt{\sum\limits_{i = 1}^{n}{{E_{i}\left( {\omega_{k},{r\; 0}} \right)} \cdot {E_{i}^{*}\left( {\omega_{k},{r\; 0}} \right)}}}}{\sqrt{\sum\limits_{i = 1}^{n}{{E_{i}\left( {\omega_{k},r} \right)} \cdot {E_{i}^{*}\left( {\omega_{k},r}\; \right)}}}}} \\{{\phi_{r}\left( \omega_{k} \right)} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}\left( {{\arg\left( {E_{i}\left( {\omega_{k},{r\; 0}} \right)} \right)} - {\arg\left( {{E_{i}\left( {\omega_{k},r} \right)}{\mathbb{e}}^{{- j}\;{{\Delta\tau}{({{t\; 0},r,i})}}\omega\; k}} \right)}} \right)}}}\end{matrix}$where:

-   Ei* is the complex conjugate value of Ei,-   and Δτ(r0,r,i)=(d(r0,i)−d(r,i))/c, d(r,i) being the distance between    the transducer i and the focusing point r, and d(r0,i) being the    distance between the transducer i and a particular focusing point    r0;    -   substep (b12) itself includes the following substeps:

(b121) p transfer matrices H(ωk)=[Hri(ωk)] are determined, i rangingfrom 1 to n and r ranging from 1 to m, where Hri(ωk) is the value, atthe frequency ωk, of the Fourier transform of the impulse responsehri(t),

(b122′) the transfer matrices H(ωk) are corrected in order to overcomethe aberrations generated by the aberrating wall in the vicinity of eachfocusing point r, this correction being carried out on the basis of theimpulse responses hri(t) determined previously, and corrected transfermatrices H′(ωk) are obtained in this way,

(b123′) for each focusing point r corresponding to a transducer of thetarget array, n components E′i(ωk,r) are determined, i varying between 1and n, such that F(ωk,r)=H′(ωk).E′(ωk,r), where E′(ωk,r)=[E′i(ωk,r)] isa vector with n components, F(ωk,r) is a vector with m componentsFl(ωk,r), l varying between 1 and m, these m components Fl(ωk,r)corresponding to a desired focusing of the acoustic waves at thefrequency ωk on the focusing point r corresponding to a transducer ofthe target array,

(b124′) for each focusing point r corresponding to a transducer of thetarget array, a vector of n time signals e′(t,r)=[e′i(t,r)] is deducedtherefrom, i varying between 1 and n, where

${e_{i}^{\prime}\left( {t.r} \right)} = {\sum\limits_{k = 1}^{P}{E^{\prime}{{i\left( {{\omega\; k},r} \right)} \cdot {\mathbb{e}}^{{{j\omega}\; k},t}}}}$in complex notation, the signals e′i(t,r) being said reference signals;

-   -   p matrices H′⁻¹(ωk) are calculated during substep (b123′),        respectively by regularization and inversion of the transfer        matrices H′(ωk), and the vector E′(ωk,r) is calculated for each        transducer r of the target array by the formula:        E′(ωk,r)=H′ ⁻¹(ωk).F(ωk,j);    -   during step (b123′), the components Fl(ωk,r) of the vector        (ωk,r) corresponding to the spatial distribution of the desired        field at the frequency ωk are equal to 0 for l≠r and to 1 for        l=r;    -   during substep (b122′), the aberrating wall in the vicinity of        each focusing point r corresponding to a transducer of the        target array is assimilated to a filter, which has a finite        impulse response and is defined at each frequency ωk by an        amplitude Gr(ωk) and a phase φ_(r)(ωk), substep (b122′) itself        including the following substeps:

(b122′1) for each frequency ωk, the amplitude Gr(ωk) and the phaseφ_(r)(ωk) are calculated on the basis of the impulse responsesdetermined previously,

(b122′2) p corrected transfer matrices H′(ωk)=[H′ji(ωk)] are calculated,where

${{H_{ji}^{\prime}\left( {\omega\; k} \right)} = {{{H_{ji}\left( \omega_{k} \right)} \cdot \frac{1}{G_{j}\left( {\omega\; k} \right)}}{\mathbb{e}}^{{- j}\;{\phi_{j}{(\omega_{k})}}}}};$

-   -   during substep (b122′1), the amplitude Gr(ωk) and the phase        φ_(r)(ωk) are calculated for each frequency ωk in the following        way:

$\begin{matrix}{{{Gr}\left( \omega_{k} \right)} = \frac{\sqrt{\sum\limits_{i = 1}^{n}{{H_{r\; i}\left( \omega_{k} \right)} \cdot {H_{r\; i}^{*}\left( \omega_{k} \right)}}}}{\sqrt{\sum\limits_{i = 1}^{n}{{H_{{r\; 0},i}\left( \omega_{k} \right)} \cdot {H_{{r\; 0},i}^{*}\left( \omega_{k} \right)}}}}} \\{{{\phi_{r}\left( \omega_{k} \right)} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}\left( {{\arg\left( {{H_{r\; i}\left( \omega_{k} \right)}{\mathbb{e}}^{j\;\Delta\;{\tau{({i,r,{ro}})}}\omega\; k}} \right)} - {\arg\left( {H_{{r\; 0},i}\left( \omega_{k} \right)} \right)}} \right)}}},\mspace{14mu}{{where}:}}\end{matrix}$

-   H*ri designates the complex conjugate value of Hri,-   and Δτ(r0,r,i)=(d(r0,i)−d(r,i))/c, d(r,i) being the distance between    the transducer i and the focusing point r, and d(r0,i) being the    distance between the transducer i and a particular focusing point    r0;    -   during step (c), substep (c1) is followed by the following        substeps:

(c2) said transducers of the imaging array are made to pick up signalss_(i)(t) back-scattered by the dissipative heterogeneous medium,

(c3) the reference signal emitted by each transducer of the imagingarray is convoluted with the back-scattered signal picked up by thistransducer,

(c4) then the convolution products obtained in this way are summed,

step (c) being repeated for a plurality of points lying in thesubstantially homogeneous medium;

-   -   during substep (b21), the first estimate of each reference        signal is e′i(t,q)=e′i(ts+θi(q),q0) for each focusing point q,        q0 being the index of a focusing point close to the focusing        point q for which the reference signal has already been        determined, θi(q) being a delay equal to a value δi(q)/c, where        c is the average speed of the acoustic waves in the medium, and        δi(q) is equal to a difference between, on the one hand, a        distance between the transducer i of the imaging array and the        focusing point q0, and, on the other hand, a distance between        the transducer i of the imaging array and the focusing point q;    -   during substep (b2), when at least certain transducers with        index v of the imaging array are not directly in contact with        the aberrating layer, the corresponding signals e′_(v)(t,q) are        corrected by digital backpropagation in order to simulate        transducers placed in direct contact with the aberrating layer;    -   during substep (b22), the values α_(i)(q) and τ_(i)(q) are        looked for to maximize the following coherence criterion C:

${C = \frac{< \left| {\sum\limits_{i = 1}^{n}{\alpha_{i} \cdot {g_{i}\left( {{t - \tau_{i}},q} \right)}}} \middle| {}_{2} > \right.}{\left. {{n \cdot \sum\limits_{i = 1}^{n}} <} \middle| {\alpha_{i} \cdot {g_{i}\left( {{t - \tau_{i}},q} \right)}} \middle| {}_{2} > \right.}},\mspace{14mu}{{where}:}$

-   g_(i)(t,q)=s_(i)(t){circle around (x)}e_(i)(t,q), {circumflex over    (x)} representing the convolution operation,-   and <> represents a time average;    -   during substep (b22), the values τ_(i)(q) are calculated by        maximizing a cross-correlation function, for transducers close        to the imaging array, of the signals g_(i)(t,q) and        g_(i+1)(t,q);    -   during substep (b22), the values α_(i)(q) are calculated so as        to equalize the maximum amplitude of the functions g_(i)(t,q) on        the index i;    -   during substep (b22), the values α_(i)(q) and τ_(i)(q) are        calculated by carrying out a cross-correlation, for transducers        close to the imaging array, of the signals g_(i)(t,q) and        g_(i+1)(t,q);    -   during substep (b22), the values α_(i)(q) and τ_(i)(q) are        calculated so as to equalize the maximum amplitude of the        functions g_(i)(t,q) on the index i;    -   substep (b22) relating to each focusing point q is carried out        immediately after substep (b21) relating to the same focusing        point q;    -   the dissipative heterogeneous medium consists of the brain        surrounded by the skull;    -   the imaging array and the target array are two separate arrays        arranged on either side of the dissipative heterogeneous medium;    -   all the transducers belong both to the imaging array and to the        target array;    -   the acoustic waves are ultrasound waves.

The invention furthermore relates to a device designed for implementingthe method defined above.

Other characteristics and advantages of the invention will becomeapparent during the following description of one of its embodiments,which is given by way of nonlimiting example with reference to theappended drawing.

BRIEF DESCRIPTION OF THE DRAWINGS

In the drawing, the single FIGURE represents an ultrasound imagingdevice according to one embodiment of the invention.

DETAILED DESCRIPTION OF THE DISCLOSURE

The ultrasound imaging device 1 represented in the drawing is designedto produce an ultrasound echographic image of a patient's brain 2 (atfrequencies of, for example, the order of from 1 to 3 MHz) from outsidethe skull 3, the brain 2 constituting a substantially homogeneous mediumfor propagation of the acoustic waves and the skull 3 constituting adissipative aberrating layer, so that the overall cranium 2, 3constitutes a dissipative heterogeneous medium.

As a variant, the invention could be applicable, in particular:

-   -   to acoustic imaging of any other nonhomogeneous dissipative        heterogeneous medium comprising a substantially homogeneous        medium surrounded by a relatively thin dissipative layer        generating aberrations in the propagation of the ultrasound        waves,    -   or to any other method involving at least one focusing upon        emission into such a medium.

In the example represented in the drawing, the imaging device 1 includesa microcomputer 4, or any other device for control and/or visualizationof ultrasound images, this microcomputer conventionally including akeyboard 4 a, optionally combined with other control interfaces, and ascreen 4 b making it possible to visualize the images of the brain 2.

The imaging device 1 furthermore includes two arrays 5, 6 of ultrasoundtransducers T1, T2 . . . Tn and T′1, T′2 . . . T′m forming, for example,two linear banks of transducers which are arranged on either side of theuser's skull 3, in predetermined geometrical positions with respect toone another, each transducer bank 5, 6 being brought into contact withthe skull 3 via a layer 7 of gel or the like.

The various transducers T1, T2 . . . Tn and T′1, T′2 . . . T′m may becontrolled directly by the microcomputer 4, or preferably by a centralelectronic unit CPU which is contained, for example, in an electronicsrack 8 and is itself controlled by the microcomputer 4.

Advantageously, each of the transducers T1, T2 . . . Tn, T′1, T′2, T′mis connected to a sampler, respectively E1, E2 . . . En, E′1, E′2, E′m,and each sampler is itself connected to a memory, respectively M1, M2 .. . Mm, M′1, M′2 . . . M′m and to a central unit C1, C2 . . . Cm, C′2,C′2 . . . C′m. These memories and these central units are in turnconnected, directly or indirectly, to the aforementioned central unitCPU, which is furthermore connected to at least one central memory M.

The device which has just been described operates as follows.

Initially, the two arrays of transducers 5, 6 are fixed on either sideof the patient's skull 3, in said predetermined positions. To this end,the arrays of transducers 5, 6, respectively referred to as the imagingarray and the target array, may be carried by a rigid support such as ahat (not shown) arranged around the patient's head.

The device then follows a learning step lasting a few minutes(advantageously from 1 to 3 min), making it possible to take account ofall the propagation aberrations due to the nonhomogeneous nature of thedissipative medium formed by the skull 3 and the brain 2.

During this learning step, firstly each of the transducers T1, T2 . . .Ti, . . . Tn, of the imaging array 5 is made to successively emit anacoustic pulse, and, for each pulse emitted by one of the transducers Tiof the imaging array, the signal picked up by the transducers T′1, T′1 .. . T′r, . . . T′m of the target array 6 is recorded, that is to say theimpulse response hri(t) of the dissipative heterogeneous medium betweenthe transducer i in question of the imaging array 5 and each transducerj of the target array 6.

Each impulse response hri(t) is recorded in digital form with a certaintime sampling which determines a certain number p of monochromaticfrequency components of the impulse response, each corresponding to afrequency ωk, k being an index between 1 and p.

In the case envisaged here, where at least certain transducers of thetarget array and/or of the imaging array are not directly in contactwith said aberrating layer 3, the impulse responses are corrected inorder to simulate virtual transducers arranged in contact with saidaberrating layer. The position of the layer with respect to thetransducers may optionally be obtained by conventional imaging(ultrasound echography, x-ray scanner, MRI, etc.). The corrected impulseresponses are calculated by a known digital backpropagation algorithm,described in particular in the following articles:

-   -   “Ultrasonic beam steering through inhomogeneous layers with a        time reversal mirror”, C. DORME, M. FINK, IEEE Transactions        Ultrasonics, Ferroelectric and Frequency Control, 43 (1),        January 1996, pp. 167–175,    -   “Focusing and steering through absorbing and aberrating layers:        Application to ultrasonic propagation through the skull” Journal        of Acoustical Society of America, 103 (5), May 1998, pp.        2403–2410,    -   and “Propagation and backpropagation for ultrasonic wavefront        design” Liu, D.-L., and Waag, R. C. IEEE Trans. on Ultras.        Ferro. and Freq. Contr. 44(1):1–13 (1997).

In what follows, hri(t) will therefore denote the impulse responses for(real or virtual) elements lying against the aberrating layer. Thevirtual or real elements lying against the aberrating layer 3 willfurthermore be referred to below as “focusing points” of index r between1 and m.

When the transducers of the imaging array 5 emit acoustic signalse_(i)(t), these signals generate acoustic signals fr(t) expressed asfollows at the transducers r of the target array 6:

${{{fr}(t)} = {\sum\limits_{i = 1}^{n}{{{hri}(t)} \otimes {{ei}(t)}}}},$where {circumflex over (x)} represents the time convolution operation.

After Fourier transform, this equation becomes:F(ωk)=H(ωk).E(ωk), where:

-   -   H(ωk) is the transfer matrix, of size m*n, between the        transducers Ti of the imaging array and the transducers Tr of        the target array: the components Hri(ωk) of this matrix are the        components of the Fourier transforms of the impulse responses        hri(t) at the frequency ωk,    -   E(ωk) is a vector whose components E_(i)(ωk) are the components        of the Fourier transform of the aforementioned signals e_(i)(t)        at the frequency ωk,    -   and F(ωk) is a vector whose components F_(j)(ωk) are the        components of the Fourier transform of the aforementioned        signals f_(j)(t) at the frequency ωk.

By inverting each transfer matrix H(ωk), it is therefore possible todetermine the vector E(ωk,j) which is suitable for generating, at thefocusing point r corresponding to the transducer T′r of the targetarray, a vector F(ωk,j) all of whose components are as close as possibleto the objective initially fixed (preferably all equal to zero, exceptfor the component with index j corresponding to the transducer T′j,which is equal to 1 when the intention is to emit an acoustic pulse atthe focusing point r), by virtue of the relationship:E(ωk,j)=H ⁻¹(ωk).F(ωk,j),where H⁻¹(ωk) is the inverse matrix of H(ωk).

H⁻¹(ωk) may be calculated by singular value decomposition, for example,which makes it possible to regularize the inversion of the matrix H(ωk).

Next, by inverse Fourier transform of the various components Ei(ωk,j) ofthe vector E(ωk,j), the various reference signals ei(t,j) are determinedwhich are suitable for focusing an acoustic pulse (or optionally anotheracoustic signal) at the focusing point r, when they are emitted by thevarious transducers Ti of the imaging array 5. Focusing of the imagingarray 5 on each transducer of the target array 5 is therefore carriedout by an inverse spatiotemporal filter.

The central unit CPU then follows a process of learning the aberrationsat the target array due to the wall of the skull 3.

During this process, these aberrations are considered as a filter with afinite impulse response.

In the Fourier domain, this filter is defined at each frequency ωk by anamplitude Gr(ωk) and a phase φ_(r)(ωk).

In order to calculate these coefficients, the phase and the amplitude ofall the vectors Er are compared. To this end, the first stage is toeliminate the phase shifts introduced by the path differences betweenthe imaging transducers Ti and the various focusing points indexed r.This is equivalent to selecting a particular focusing point r0 andintroducing a linear phase shift for the others with the angularfrequency: exp(−jΔτ(ro,r,1)ω) with Δτ(r0,r,i)=(d(r0,i)−d(r,i))/c whered(r,i) is the distance between the transducer i and the focusing pointr, and c is the average speed of the acoustic waves in the medium to beimaged, in this case the brain 2.

Once this correction has been carried out, the differences in amplitudeand phase between the vectors Er are attributed to the aberrating layer3 which lie against the target array. The gain factor Gj(ωk) and thephase factor φ_(j)(ωk) are then calculated for each focusing point r:

$\begin{matrix}{{{Gr}\left( \omega_{k} \right)} = \frac{\sqrt{\sum\limits_{i = 1}^{n}{{E_{i}\left( {\omega_{k},{r\; 0}} \right)} \cdot {E_{i}^{*}\left( {\omega_{k},{r\; 0}} \right)}}}}{\sqrt{\sum\limits_{i = 1}^{n}{{E_{i}\left( {\omega_{k},r} \right)} \cdot {E_{i}^{*}\left( {\omega_{k},r}\; \right)}}}}} \\{{\phi_{r}\left( \omega_{k} \right)} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}\left( {{\arg\left( {E_{i}\left( {\omega_{k},{r\; 0}} \right)} \right)} - {\arg\left( {{E_{i}\left( {\omega_{k},r} \right)}{\mathbb{e}}^{{- j}\;{{\Delta\tau}{({{t\; 0},r,i})}}\omega\; k}} \right)}} \right)}}}\end{matrix}$where Ei* is the complex conjugate value of Ei.

The pairs {Gj(ωk), φ_(j)(ωk)} correspond to the relative attenuationfactor and relative phase shift introduced at each frequency by theportion of the aberrating layer 3 lying against the focusing point r.They therefore finally characterize the aberrations introduced by theaberrating layer portion lying against the target array.

The aberrations introduced by the aberrating layer 3 lying against thetarget transducers are then eliminated in all the p matricesH(ωk)=[Hji(ωk)] defined above.

To this end, a new set of transfer matrices H′(ωk)=[H′ji(ωk)] iscalculated characterizing the propagation between the imaging array andthe target array in a virtual medium for which only the aberrationslying against the imaging array remain:

${H_{ji}^{\prime}\left( \omega_{k} \right)} = {{{H_{ji}\left( \omega_{k} \right)} \cdot \frac{1}{G_{j}\left( \omega_{k} \right)}}{{\mathbb{e}}^{{- j}\;{\phi_{j}{(\omega_{k})}}}.}}$

For each transducer r of the target array, n components E′i(ωk,r) arethen determined, i varying between 1 and n, such thatF(ωk,r)=H′(ωk).E′(ωk,r), where E′(ωk,r)=[E′i(ωk,r)] is a vector with ncomponents, F(ωk,r) is a vector with m components Fl(ωk,r), l varyingbetween 1 and m, these m components Fl(ωk,r) corresponding to a desiredfocusing of the acoustic waves at the frequency ωk on the focusing pointr corresponding to a transducer of the target array.

For each focusing point r corresponding to a transducer of the targetarray, a vector of n reference time signals e′(t,r)=[e′i(t,r)] isdeduced therefrom, i varying between 1 and n, where

${e_{i}^{\prime}\left( {t,r} \right)} = {\sum\limits_{k = 1}^{P}{E\;{{i\left( {{\omega\; k},r} \right)} \cdot {\mathbb{e}}^{{j\;\omega\; k},t}}}}$in complex notation.

These reference signals e′i(t,j) are adapted so that the emission ofthem respectively by the various transducers i of the imaging arraygenerates an acoustic pulse focused on the transducer j of the targetarray in the absence of the aberrating layer lying against the targetarray.

It will be noted that, as a variant, the reference signals could bedetermined in the following way, after having determined the impulseresponses hri(t) and the p transfer matrices H(ωk):

-   -   the transfer matrices H(ωk) are corrected in order to overcome        the aberrations generated by the aberrating wall 3 in the        vicinity of each focusing point r, this correction being carried        out on the basis of the impulse responses hri(t) determined        previously, and corrected transfer matrices H′(ωk) are obtained        in this way,    -   by inverting the matrices H′(ωk), for each focusing point r        corresponding to a transducer of the target array, n components        E′i(ωk,r) are determined, i varying between 1 and n, such that        F(ωk,r)=H′(ωk).E′(ωk,r), where E′(ωk,r)=[E′i(ωk,r)] is a vector        with n components, F(ωk,r) is a vector with m components        Fl(ωk,r), l varying between 1 and m, these m components Fl(ωk,r)        corresponding to a desired focusing of the acoustic waves at the        frequency ωk on the focusing point r corresponding to a        transducer of the target array,    -   and for each focusing point r corresponding to a transducer of        the target array, a vector of n time signals e′(t,r)=[e′i(t,r)]        is deduced therefrom, i varying between 1 and n, where

${e_{i}^{\prime}\left( {t,r} \right)} = {\sum\limits_{k = 1}^{P}{E\;{{i\left( {{\omega\; k},r} \right)} \cdot {\mathbb{e}}^{{j\;\omega\; k},t}}}}$in complex notation, the signals e′i(t,r) being said reference signals.

Advantageously, during the calculation of the matrices H′(ωk), theaberrating wall in the vicinity of each focusing point r correspondingto a transducer of the target array is assimilated to a filter, whichhas a finite impulse response and is defined at each frequency ωk by anamplitude Gr(ωk) and a phase φ_(r)(ωk), which are calculated as follows:

$\begin{matrix}{{G_{r}\left( \omega_{k} \right)} = \frac{\sqrt{\sum\limits_{i = 1}^{n}{{H_{r\; i}\left( \omega_{k} \right)} \cdot {H_{r\; i}^{*}\left( \omega_{k} \right)}}}}{\sqrt{\sum\limits_{i = 1}^{n}{{H_{{r\; 0},i}\left( \omega_{k} \right)} \cdot {H_{{r\; 0},i}^{*}\left( \omega_{k} \right)}}}}} \\{{{\phi_{r}\left( \omega_{k} \right)} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}\left( {{\arg\left( {{H_{ri}\left( {\omega\;}_{k} \right)}{\mathbb{e}}^{j\;\Delta\;{\tau{({i,r,{ro}})}}\omega\; k}} \right)} - {\arg\left( {H_{{r\; 0},i}\left( \omega_{k} \right)} \right)}} \right)}}},{{where}\text{:}}}\end{matrix}$

-   H*ri designates the complex conjugate value of Hri,-   and Δτ(r0,r,i)=(d(r0,i)−d(r,i))/c, d(r,i) being the distance between    the transducer i and the focusing point r, and d(r0,i) being the    distance between the transducer i and a particular focusing point    r0.

p corrected transfer matrices H′(ωk)=[H′ji(ωk)] are then calculated,where

${{H_{ji}^{\prime}\left( \omega_{k} \right)} = {{{H_{ji}\left( \omega_{k} \right)} \cdot \frac{1}{G_{j}\left( \omega_{k} \right)}}{\mathbb{e}}^{{- j}\;{\phi_{j}{(\omega_{k})}}}}},$which are used to determine the vectors E′i(ωk) as explained above, andtherefore the various reference signals e′(t,r), r ranging from 1 to m.

The central unit CPU then learns to focus at a number R of predeterminedfocusing points lying in the brain 2, with indices q between m+1 andm+R, this substep consisting in determining for each of these focusingpoints q, moving step-by-step away from the transducers of the targetarray, reference signals e′i(t,q) to be emitted by the varioustransducers of the imaging array in order to generate a pulse focused onsaid focusing point q.

The reference signals e′i(t,q) are initially determined, for each newfocusing point q, in the form e′i(t,q)=e′i(t+θi(q),q0) for each focusingpoint q, q0 being the index of a focusing point close to the focusingpoint q for which the reference signal has already been determined, thedelay θi(q) initially being equal to a value δi(q)/c, where c is theaverage speed of the acoustic waves in the medium, and δi(q) is equal toa difference between, on the one hand, a distance between the transduceri of the imaging array and the focusing point q0, and, on the otherhand, a distance between the transducer i of the imaging array and thefocusing point q.

In the event that certain transducers with index v of the imaging arraydo not lie against the aberrating layer, it is furthermore desirable tocorrect the reference signals the corresponding signals e′_(v)(t,q) bydigital backpropagation from the virtual transducers (lying against theaberrating layer 3) to the real transducers (separated from said layer 3by some gel 7 or the like), in a manner which is known per se, by themethod which is the reverse of that described above in relation to theimpulse responses.

The transducers of the imaging array are then made to emit, byiterations, the estimates obtained for the reference signals e′i(t,q),then signals s_(i)(t,q) back-scattered by the dissipative heterogeneousmedium are picked up with the same transducers.

Next, these reference signals e′i(t,q) are modified for the nextiteration in the following way:e_(i)′(t)→α_(i)(q).e_(i)′(t−τ_(i)(q))where the values τ_(i)(q) and α_(i)(q) are a corrective delay and acorrective amplitude factor, which are calculated so as to maximize acoherence criterion C between said back-scattered signals.

Advantageously, this coherence criterion C may be the following:

${C = \frac{< \left| {\sum\limits_{i = 1}^{n}{\alpha_{i} \cdot {g_{i}\left( {{t - \tau_{i}},q} \right)}}} \middle| {}_{2} > \right.}{\left. {{n \cdot \sum\limits_{i = 1}^{n}} <} \middle| {\alpha_{i} \cdot {g_{i}\left( {{t - \tau_{i}},q} \right)}} \middle| {}_{2} > \right.}},{{where}:}$

-   g_(i)(t,q)=s_(i)(t){circumflex over (x)}e_(i)′(t,q) {circumflex over    (x)} representing the convolution operation,-   and <> represents a time average.

In this optimization process, the values τ_(i)(q) may be calculated ateach iteration so as to maximize a cross-correlation function, fortransducers close to the imaging array, of the aforementioned signalsg_(i)(t,q) and g_(i+1)(t,q), and the values α_(i)(q) may be calculatedso as to equalize the maximum amplitude of the functions g_(i)(t,q) onthe index i.

The reference signals e′i(t,q), i ranging from 1 to n, are henceoptimized so that they produce an acoustic signal focused precisely onthe focusing point q lying in the brain. This optimization process hasalready been explained in more detail by Mallart et al. (The VanCittert-Zernike theorem in pulse echo measurements, J. Acoust. Soc. Am.90(5), November 1991, pp. 2716–2727; Adaptive focusing in scatteringmedia through sound speed inhomogeneities: the Van Cittert Zernikeapproach and focusing criterion, J. Acoust. Soc. Am. 96(6), December1994, pp. 3721–3732).

When this optimization is completed for a focusing point q, for exampleafter 2 or 3 iterations when the criterion C has reached a predeterminedvalue (in particular close to ⅔), operation proceeds to the nextfocusing point q+1, etc.

The reference signals e′i(t,q) obtained in this way are stored, forexample in the memories M1–Mn.

Once the learning step is completed, it is in particular possible toproduce echographic images of the brain 2, optionally at a fast ratewhich may be as high as the speed of a standard echograph, for example20 to 30 images per second. In order to produce each of these images,the following procedure is adopted for each focusing point q belongingto the field to be observed:

-   -   the transducers Ti of the imaging array are respectively made to        emit said reference signals ei(t,q),    -   then said transducers of the imaging array are made to pick up        signals si(t) back-scattered by the viscoelastic medium,    -   the reference signal ei(t,q) emitted by each transducer of the        imaging array is convoluted with the back-scattered signal si(t)        picked up by this transducer,    -   then the convolution products obtained in this way are summed.

It will be noted that the various aforementioned operations carried outduring the learning step or the imaging step may either be programmed inthe central unit CPU, or all or some of them may be performed byspecialized circuits.

Furthermore, it will also be noted that all the transducers Ti, T′rcould be used to produce the echographic images of the brain. In thiscase, the imaging array would be the same as the target array, and eachof these two arrays would comprise all the transducers, the operationdescribed above then being applied mutatis mutandis.

1. A noninvasive method for focusing acoustic waves in a dissipativeheterogeneous medium (2, 3) comprising a substantially homogeneousmedium (2) surrounded at least partially by a dissipative aberratinglayer (3) which generates aberrations in the propagation of the acousticwaves, the acoustic waves being emitted from outside the aberratinglayer (3) and focused in the substantially homogeneous medium (2),characterized in that it includes the following steps: (a) an initialpositioning step during which a number t greater than 2 of acoustictransducers (T1–Tn, T′1–T′m) are fixed in predetermined positionsoutside the aberrating layer (3), these transducers being in contactwith said aberrating layer and forming at least: an imaging array(T1–Tn) which combines a number n between 1 and t of said transducers,and a target array (T′1–T′m) which combines a number m between 1 and tof said transducers, (b) a learning step itself comprising the followingsubsteps: (b1) a substep of learning to focus the imaging array on thetarget array, during which substep: (b11) impulse responses hri(t) ofthe dissipative heterogeneous medium are determined, respectivelybetween each transducer i of the imaging array and a plurality offocusing points r lying on the aberrating layer (3) in respectivecorrespondence with transducers of the target array, these impulseresponses being stored in digital form with a certain time samplingwhich determines a number p of frequency components of the impulseresponse, with respective frequencies ωk, i being an index between 1 andn which designates a transducer of the imaging array, r being an indexbetween 1 and m which designates a focusing point corresponding to atransducer of the target array and k being an index between 1 and pwhich designates a frequency component, (b12) on the basis of theseimpulse responses, for each focusing point r corresponding to atransducer of the imaging array, a set of n reference time signalse′i(t,r) is calculated, i varying between 1 and n, such that if theaberrating wall were removed in the vicinity of the focusing point r,the emission of these reference signals by the various transducers i ofthe imaging array would generate a predetermined signal focused on thefocusing point r, (b2) a substep of focusing at a number R ofpredetermined focusing points lying in the substantially homogeneousmedium, with indices q between m+1 and m+R, this substep consisting indetermining for each of these focusing points q, moving step-by-stepaway from the focusing points 1 to m corresponding to the transducers ofthe target array, reference signals e′i(t,q) to be emitted by thevarious transducers i of the imaging array in order to generate a pulsefocused on said focusing point q, the reference signals e′i(t,q) beingdetermined for each focusing point q by proceeding as follows: (b21) afirst estimate of e′i(t,q), for i ranging from 1 to n, is calculated onthe basis of at least one reference signal e′i(t,q0), q0 being the indexof at least one focusing point close to the focusing point q for whichthe reference signal has already been determined, this calculation beingperformed by using an average speed of the acoustic waves in thesubstantially homogeneous medium (2), (b22) the transducers of theimaging array are made to emit, by iterations, the estimates previouslyobtained of the reference signals e′i(t,q), then signals s_(i)(t,q)back-scattered by the dissipative heterogeneous medium are picked upwith the same transducers, then these reference signals e′i(t,q) aremodified for the next iteration in the following way:e _(i)′(t)→α _(i)(q).e _(i)′(t−τ _(i)(q))  where the values α_(i)(q) andτ_(i)(q) are a corrective amplitude factor and a corrective delay, whichare calculated so as to maximize a coherence criterion C between saidback-scattered signals, said iterations being stopped when the criterionC reaches a predetermined threshold, (b3) the reference signals e′i(t,q)are stored, at least for q between m+1 and m+R, (c) and a focusing stepduring which, for at least one of said focusing points q, thetransducers of the imaging array are made to emit said reference signalse′_(i)(t,q), i being an index between 1 and n designating a transducerof the imaging array.
 2. The method as claimed in claim 1, in whichduring substep (b11), when at least certain transducers (T1–Tm, T′1–T′m)are in contact with an intermediate heterogeneous medium which is itselfin contact with the aberrating layer, the impulse responses hri(t) arecorrected by digital backpropagation in order to simulate transducerslying directly in contact with the aberrating layer.
 3. The method asclaimed in claim 1 or claim 2, in which substep (b12) itself includesthe following substeps: (b121) p transfer matrices H(ωk)=[Hri(ωk)] aredetermined, i ranging from 1 to n and r ranging from 1 to m, whereHri(ωk) is the value, at the frequency ωk, of the Fourier transform ofthe impulse response hri(t), (b122) for each focusing point rcorresponding to a transducer of the target array, n components Ei(ωk,r)are determined, i varying between 1 and n, such thatF(ωk,r)=H(ωk).E(ωk,r), where E(ωk,r)=[Ei(ωk,r)] is a vector with ncomponents, F(ωk,r) is a vector with m components Fl(ωk,r), l varyingbetween 1 and m, these m components Fl(ωk,r) corresponding to a desiredfocusing of the acoustic waves at the frequency ωk on the focusing pointr corresponding to a transducer of the target array, (b123) for eachfocusing point r corresponding to a transducer of the target array, avector of n time signals e(t,r)=[ei(t,r)] is deduced therefrom, ivarying between 1 and n, where${e_{i}^{\prime}\left( {t,r} \right)} = {\sum\limits_{k = 1}^{P}{E\;{{i\left( {{\omega\; k},r} \right)} \cdot {\mathbb{e}}^{{j\;\omega\; k},t}}}}$in complex notation, these signals ei(t,r) being adapted so that theemission of them respectively by the various transducers i of theimaging array generates an acoustic pulse focused on the focusing pointr of the target array, (b124) a substep of correcting the aberrationsgenerated by the aberrating layer between the substantially homogeneousmedium and each target transducer r, these aberrations being estimatedon the basis of the measurements carried out previously, the aberrationsestimated in this way being used to calculate said reference timesignals e′i(t,r).
 4. The method as claimed in claim 3, in which pmatrices H⁻¹(ωk) are calculated during substep (b122), respectively byregularization and inversion of the transfer matrices H(ωk), and thevector E(ωk,r) is calculated for each transducer r of the target arrayby the formula:E(ωk,r)=H ⁻¹(ωk).F(ωk,j).
 5. The method as claimed in claim 3 or claim4, in which during step (b122), the components Fl(ωk,r) of the vectorF(ωk,r) corresponding to the spatial distribution of the desired fieldat the frequency ωk are equal to 0 for l≠r and to 1 for l=r.
 6. Themethod as claimed in any one of claims 3 to 5, in which during substep(b124), the aberrating wall in the vicinity of each focusing point rcorresponding to a transducer of the target array is assimilated to afilter, which has a finite impulse response and is defined at eachfrequency ωk by an amplitude Gr(ωk) and a phase φ_(r)(ωk), substep(b124) itself including the following substeps: (b1241) for eachfrequency ok, the amplitude Gr(ωk) and the phase φ_(r)(ωk) arecalculated on the basis either of the signals ei(t,r) or of the vectorsE(ωk,r), (b1242) p corrected transfer matrices H′(ωk)=[H′ji(ωk)] arecalculated, where${{H_{ji}^{\prime}\left( \omega_{k} \right)} = {{{H_{ji}\left( \omega_{k} \right)} \cdot \frac{1}{G_{j}\left( \omega_{k} \right)}}{\mathbb{e}}^{{- j}\;{\phi_{j}{(\omega_{k})}}}}},$(b1243) for each transducer r of the target array, n componentsE′i(ωk,r) are determined, i varying between 1 and n, such thatF(ωk,r)=H′(ωk).E′(ωk,r), where E′(ωk,r)=[E′i(ωk,r)] is a vector with ncomponents, F(ωk,r) is a vector with m components Fl(ωk,r), l varyingbetween 1 and m, these m components Fl(ωk,r) corresponding to a desiredfocusing of the acoustic waves at the frequency ωk on the focusing pointr corresponding to a transducer of the target array, (b1244) for eachfocusing point r corresponding to a transducer of the target array, avector of n reference time signals e′(t,r)=[e′i(t,r)] is deducedtherefrom, i varying between 1 and n, where${e_{i}^{\prime}\left( {t,r} \right)} = {\sum\limits_{k = 1}^{P}{E\;{{i\left( {{\omega\; k},r} \right)} \cdot {\mathbb{e}}^{{j\;\omega\; k},t}}}}$in complex notation.
 7. The method as claimed in claim 6, in whichduring substep (b1241), the amplitude Gr(ωk) and the phase φ_(r)(ωk) arecalculated as follows: $\begin{matrix}{{{Gr}\left( \omega_{k} \right)} = \frac{\sqrt{\sum\limits_{i = 1}^{n}{{E_{i}\left( {\omega_{k},{r\; 0}} \right)} \cdot {E_{i}^{*}\left( {\omega_{k},{r\; 0}} \right)}}}}{\sqrt{\sum\limits_{i = 1}^{n}{{E_{i}\left( {\omega_{k},r} \right)} \cdot {E_{i}^{*}\left( {\omega_{k},r} \right)}}}}} \\{{\phi_{r}\left( \omega_{k} \right)} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}\left( {{\arg\left( {E_{i}\left( {\omega_{k},{r\; 0}} \right)} \right)} - {\arg\left( {{E_{i}\left( {\omega_{k},r} \right)}{\mathbb{e}}^{{- j}\;{{\Delta\tau}{({{r\; 0},r,i})}}\omega\; k}} \right)}} \right)}}}\end{matrix}$ where: Ei* is the complex conjugate value of Ei, andΔτ(r0,r,i)=(d(r0,i)−d(r,i))/c, d(r,i) being the distance between thetransducer i and the focusing point r, and d(r0,i) being the distancebetween the transducer i and a particular focusing point r0.
 8. Themethod as claimed in claim 1 or claim 2, in which substep (b12) itselfincludes the following substeps: (b121) p transfer matricesH(ωk)=[Hri(ωk)] are determined, i ranging from 1 to n and r ranging from1 to m, where Hri(ωk) is the value, at the frequency ωk, of the Fouriertransform of the impulse response hri(t), (b122′) the transfer matricesH(ωk) are corrected in order to overcome the aberrations generated bythe aberrating wall in the vicinity of each focusing point r, thiscorrection being carried out on the basis of the impulse responseshri(t) determined previously, and corrected transfer matrices H′(ωk) areobtained in this way, (b123′) for each focusing point r corresponding toa transducer of the target array, n components E′i(ωk,r) are determined,i varying between 1 and n, such that F(ωk,r)=H′(ωk).E′(ωk,r), whereE′(ωk,r)=[E′i(ωk,r)] is a vector with n components, F(ωk,r) is a vectorwith m components Fl(ωk,r), l varying between 1 and m, these mcomponents Fl(ωk,r) corresponding to a desired focusing of the acousticwaves at the frequency ωk on the focusing point r corresponding to atransducer of the target array, (b124′) for each focusing point rcorresponding to a transducer of the target array, a vector of n timesignals e′(t,r)=[e′i(t,r)] is deduced therefrom, i varying between 1 andn, where${{\mathbb{e}}_{i}^{\prime}\left( {t,r} \right)} = {\sum\limits_{k = 1}^{p}\;{E^{\prime}{{i\left( {{\omega\; k},r} \right)} \cdot {\mathbb{e}}^{{j\omega}\;{k.t}}}}}$in complex notation, the signals e′i(t,r) being said reference signals.9. The method as claimed in claim 8, in which p matrices H′⁻¹(ωk) arecalculated during substep (b123′), respectively by regularization andinversion of the transfer matrices H′(ωk), and the vector E′(ωk,r) iscalculated for each transducer r of the target array by the formula:E′(ωk,r)=H′ ⁻¹(ωk).F(ωk,j).
 10. The method as claimed in claim 8 orclaim 9, in which during step (b123′), the components Fl(ωk,r) of thevector F(ωk,r) corresponding to the spatial distribution of the desiredfield at the frequency ok are equal to 0 for l≠r and to 1 for l=r. 11.The method as claimed in any one of claims 8 to 10, in which duringsubstep (b122′), the aberrating wall in the vicinity of each focusingpoint r corresponding to a transducer of the target array is assimilatedto a filter, which has a finite impulse response and is defined at eachfrequency ωk by an amplitude Gr(ωk) and a phase φ_(r)(ωk), substep(b122′) itself including the following substeps: (b122′1) for eachfrequency ωk, the amplitude Gr(ωk) and the phase φ_(r)(ωk) arecalculated on the basis of the impulse responses determined previously,(b122′2) p corrected transfer matrices H′(ωk)=[H′ji(ωk)] are calculated,where${H_{ji}^{\prime}\left( \omega_{k} \right)} = {{{H_{ji}\left( \omega_{k} \right)} \cdot \frac{1}{G_{j}\left( \omega_{k} \right)}}{{\mathbb{e}}^{- {{j\phi}_{j}{(\omega_{k})}}}.}}$.
 12. The method as claimed in claim 11, in which during substep(b122′1), the amplitude Gr(ωk) and the phase φ_(r)(ωk) are calculatedfor each frequency ωk in the following way:${G_{r}\left( \omega_{k} \right)} = \frac{\sqrt{\sum\limits_{i = 1}^{n}\;{{H_{ri}\left( \omega_{k} \right)} \cdot {H_{ri}^{*}\left( \omega_{k} \right)}}}}{\sqrt{\sum\limits_{i = 1}^{n}\;{{H_{{r\; 0},i}\left( \omega_{k} \right)} \cdot {H_{{r0},i}^{*}\left( \omega_{k} \right)}}}}$${{\phi_{r}\left( \omega_{k} \right)} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}\;\left( {{\arg\left( {{H_{ri}\left( \omega_{k} \right)}{\mathbb{e}}^{{j\Delta}\;{r{({i,r,{ro}})}}\omega_{k}}} \right)} - {\arg\left( {H_{{r0},i}\left( \omega_{k} \right)} \right)}} \right)}}},{{where}\text{:}}$H*ri designates the complex conjugate value of Hri, andΔτ(r0,r,i)=(d(r0,i)−d(r,i))/c, d(r,i) being the distance between thetransducer i and the focusing point r, and d(r0,i) being the distancebetween the transducer i and a particular focusing point r0.
 13. Themethod as claimed in any one of the preceding claims, in which duringstep (c), substep (c1) is followed by the following substeps: (c2) saidtransducers of the imaging array are made to pick up signals s_(i)(t)back-scattered by the dissipative heterogeneous medium, (c3) thereference signal emitted by each transducer of the imaging array isconvoluted with the back-scattered signal picked up by this transducer,(c4) then the convolution products obtained in this way are summed, step(c) being repeated for a plurality of points lying in the substantiallyhomogeneous medium.
 14. The method as claimed in any one of thepreceding claims, in which during substep (b21), the first estimate ofeach reference signal is e′i(t,q)=e′i(ts+θi(q),q0) for each focusingpoint q, q0 being the index of a focusing point close to the focusingpoint q for which the reference signal has already been determined,θi(q) being a delay equal to a value δi(q)/c, where c is the averagespeed of the acoustic waves in the medium, and δi(q) is equal to adifference between, on the one hand, a distance between the transducer iof the imaging array and the focusing point q0, and, on the other hand,a distance between the transducer i of the imaging array and thefocusing point q.
 15. The method as claimed in any one of the precedingclaims, in which during substep (b2), when at least certain transducerswith index v of the imaging array are not directly in contact with theaberrating layer, the corresponding signals e′_(v)(t,q) are corrected bydigital backpropagation in order to simulate transducers placed indirect contact with the aberrating layer.
 16. The method as claimed inany one of the preceding claims, in which during substep (b22), thevalues α_(i)(q) and τ_(i)(q) are looked for to maximize the followingcoherence criterion C:${C = \frac{< {{\sum\limits_{i = 1}^{n}\;{\alpha_{i} \cdot {g_{i}\left( {t - {\tau_{i}q}} \right)}}}}^{2} >}{{n \cdot \sum\limits_{i = 1}^{n}}\; < {{\alpha_{i} \cdot {g_{i}\left( {{t - \tau_{i}},q} \right)}}}^{2} >}},{{where}\text{:}}$g_(i)(t,q)=s_(i)(t){circumflex over (x)}e′_(i)(t,q), {circumflex over(x)} representing the convolution operation, and <> represents a timeaverage.
 17. The method as claimed in claim 16, in which during substep(b22), the values τ_(i)(q) are calculated by maximizing across-correlation function, for transducers close to the imaging array,of the signals g_(i)(t,q) and g_(i+1)(t,q).
 18. The method as claimed inclaim 16 or claim 17, in which during substep (b22), the values α_(i)(q)are calculated so as to equalize the maximum amplitude of the functionsg_(i)(t,q) on the index i.
 19. The method as claimed in any one of thepreceding claims, in which substep (b22) relating to each focusing pointq is carried out immediately after substep (b21) relating to the samefocusing point q.
 20. The method as claimed in any one of the precedingclaims, in which the dissipative heterogeneous medium consists of thebrain surrounded by the skull.
 21. The method as claimed in any one ofthe preceding claims, in which: either the imaging array and the targetarray are two separate arrays arranged on either side of the dissipativeheterogeneous medium, or all the transducers belong both to the imagingarray and to the target array.
 22. The method as claimed in any one ofthe preceding claims, in which the acoustic waves are ultrasound waves.23. A device (1) designed for carrying out a method as claimed in anyone of the preceding claims, this device including a number t greaterthan 2 of acoustic transducers (T1–Tn, T′1–T′m) intended to be fixed inpredetermined positions outside the aberrating layer (3), thesetransducers being controlled by at least one central electronic unit(CPU) and forming at least: an imaging array (T1–Tn) which combines anumber n between 1 and t of said transducers, and a target array(T′1–T′m) which combines a number m between 1 and t of said transducers,the central electronic unit being designed to follow the followingsteps: (b) a learning step itself comprising the following substeps:(b1) a substep of learning to focus the imaging array on the targetarray, during which substep: (b11) impulse responses hri(t) of thedissipative heterogeneous medium are determined, respectively betweeneach transducer i of the imaging array and a plurality of focusingpoints r lying on the aberrating layer in respective correspondence withtransducers of the target array, these impulse responses being stored indigital form with a certain time sampling which determines a number p offrequency components of the impulse response, with respectivefrequencies ωk, i being an index between 1 and n which designates atransducer of the imaging array, r being an index between 1 and m whichdesignates a focusing point corresponding to a transducer of the targetarray and k being an index between 1 and p which designates a frequencycomponent, (b12) on the basis of these impulse responses, for eachfocusing point r corresponding to a transducer of the imaging array, aset of n reference time signals e′i(t,r) is calculated, i varyingbetween 1 and n, such that if the aberrating wall were removed in thevicinity of the focusing point r, the emission of these referencesignals by the various transducers i of the imaging array would generatean acoustic pulse focused on the focusing point r, (b2) a substep offocusing at a number R of predetermined focusing points lying in thesubstantially homogeneous medium, with indices q between m+1 and m+R,this substep consisting in determining for each of these focusing pointsq, moving step-by-step away from the focusing points 1 to mcorresponding to the transducers of the target array, reference signalse′i(t,q) to be emitted by the various transducers i of the imaging arrayin order to generate a predetermined signal focused on said focusingpoint q, the reference signals e′i(t,q) being determined for eachfocusing point q by proceeding as follows: (b21) a first estimate ofe′i(t,q), for i ranging from 1 to 4, is calculated on the basis of atleast one reference signal e′i(t,q0), q0 being the index of at least onefocusing point close to the focusing point q for which the referencesignal has already been determined, this calculation being performed byusing an average speed of the acoustic waves in the substantiallyhomogeneous medium (2), (b22) the transducers of the imaging array aremade to emit, by iterations, the estimates previously obtained of thereference signals e′i(t,q), then signals s_(i)(t,q) back-scattered bythe dissipative heterogeneous medium are picked up with the sametransducers, then these reference signals e′i(t,q) are modified for thenext iteration in the following way:e _(i)′(t)→α _(i)(q).e _(i)′(t−τ _(i)(q))  where the values α_(i)(q) andτ_(i)(q) are a corrective amplitude factor and a corrective delay, whichare calculated so as to maximize a coherence criterion C between saidback-scattered signals, said iterations being stopped when the criterionC reaches a predetermined threshold, (b3) the reference signals e′i(t,q)are stored, at least for q between m+1 and m+R, (c) and a focusing stepduring which, for at least one of said focusing points q, thetransducers of the imaging array are made to emit said reference signalse′i(t,q), i being an index between 1 and n designating a transducer ofthe imaging array.